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The Requirement of Total Evidence enjoins you to proportion your beliefs to the support they receive from your total evidence. How exactly are we to understand what it is asking you to do? In Knowledge and its Limits Timothy Williamson offers the following proposal. Your total evidence is just the totality of what you know. The support p receives from your total evidence is just the conditional probability of p on your entire body of knowledge. The Principle of Total Evidence thus enjoins you to have a degree of belief in p equal to the conditional probability of p on your entire body of knowledge. Williamson thinks that there is a kind of belief that comes in degrees — he calls it ‘outright belief’ — that is not to be identified with subjective probabilities. And it is degrees of outright belief that he means the Requirement of Total Evidence to constrain. This chapter discusses how Williamson's proposal is deeply flawed.

Chapter 9 examines how the proposed theory of epistemic rationality can accommodate outright beliefs, and what role such outright beliefs play in our epistemic conduct. It is argued that people need outright beliefs in addition to credences to simplify their reasoning. Outright beliefs do this by allowing thinkers to ignore small error probabilities. What is outright believed can change between contexts. When our beliefs change, we have to ask how related other beliefs, including beliefs representing uncertainties, change in light of this. It has been claimed that our beliefs change via an updating procedure resembling conditionalization. However, conditionalization is notoriously complicated. This claim is thus in tension with the explanation that the function of beliefs is to simplify our reasoning. We can resolve this puzzle by endorsing a different hypothesis about how beliefs change across contexts that better accounts for the simplifying role of beliefs.

This chapter argues for a “Platonic” conception of open-mindedness. Open-mindedness is not simply a matter of being willing to change your mind in response to a counterargument. You have to be willing to change your mind conditional on spending significant time with the argument, finding each step compelling, and being unable to expose a flaw. If you are willing to do this, then you may be open-minded toward the argument provided you also don’t violate various procedural norms and aren’t disposed to allow various affective factors to influence your beliefs (for example, you aren’t willfully ignorant). On this conception, we can explain how it is possible to hold an outright or “full” belief even while being open-minded toward arguments against that belief.

This chapter defends a semantics for epistemic modals and probability operators. This semantics is probabilistic—that is, sentences containing these expressions have sets of probability spaces as their semantic values relative to a context. Existing non-truth-conditional semantic theories of epistemic modals face serious problems when it comes to interpreting nested modal constructions such as ‘it must be possible that Jones smokes’. The semantics in this chapter solves these problems, accounting for several significant features of nested epistemic vocabulary. The chapter ends by defending a probabilistic semantics for simple sentences that do not contain any epistemic vocabulary, and by using this semantics to illuminate the relationship between credence and full belief.